We can interpret a topological group with fundamental group $\mathbb{Z}_2$ as - if we traverse any loop on this group twice, we return to the same state. Similarly, for fundamental group $\mathbb{Z}_n$, we would need to traverse a particular loop $n$ times, in order to return to the same state.
In terms of this interpretation of the fundamental group, what would having a topological group with fundamental group $\mathbb{Z}$ mean, considering $\mathbb{Z}$ is infinite?
A topological space $X$ with fundamental group $\mathbb{Z}$ is easy to imagine. $\mathbb{Z}$ is generated by just one element, so we have to think about a space $X$ where there exist a loop $\gamma$ such that all other loops are equal (homotopic) to $\gamma$ or to a product of finitely many $\gamma$ (that is, you can go arround $\gamma$ a finite number of times in two possible orientations).
Intuitively $X$ has to be a space with only ''one hole'' and $\gamma$ is a path that makes a loop aroound that hole. For example $S^1$ and $\mathbb{R}\setminus\lbrace0\rbrace$ have $\mathbb{Z}$ as fundamental groups.
In my opinion this is way easier to visualize, since there is no loop in $\mathbb{R}^3$ such that going arround two times make you return to the initial position.